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Bloch-Siegert shift in the Rabi oscillations on the "dressed" electron spin states

Ryhor Fedaruk, Agnieszka Kolasa, A. P. Saǐko

Published Jan 1, 2013• 0 citations
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Abstract

Using electron paramagnetic resonance (EPR) at Rabi frequencies, we study the Bloch-Siegert effect in the coherent dynamics of electron spin states driven by a bichromatic radiation consisting of a transverse microwave (MW) and longitudinal radio frequency (RF) fields. The Rabi oscillations on the electron spin states dressed by the MW field are observed at the double resonance, when the MW frequency equals the Larmor frequency of the spin system and the frequency ωrf of the RF field is close to the Rabi frequency ω1 in the MW field. The dispersive (|ωrf - ω1| ~ ωrf, ω1) and near-resonant (ωrf ≈ ω1) regimes are investigated. We demonstrate that the Bloch-Siegert shift increases with increas- ing RF amplitude and at the fixed RF amplitude it has the largest values in the low-frequency (ωrf < ω1) range of the dispersive regime. Experimental results were obtained in the time-resolved EPR for E' centres in quartz.

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Using electron paramagnetic resonance (EPR) at Rabi frequencies, we study the Bloch-Siegert effect in the coherent dynamics of electron spin states driven by a bichromatic radiation consisting of a transverse microwave (MW) and longitudinal radio frequency (RF) fields. The Rabi oscillations on the electron spin states dressed by the MW field are observed at the double resonance, when the MW frequency equals the Larmor frequency of the spin system and the frequency ωrf of the RF field is close to the Rabi frequency ω1 in the MW field. The dispersive (|ωrf - ω1| ~ ωrf, ω1) and near-resonant (ωrf ≈ ω1) regimes are investigated. We demonstrate that the Bloch-Siegert shift increases with increas- ing RF amplitude and at the fixed RF amplitude it has the largest values in the low-frequency (ωrf < ω1) range of the dispersive regime. Experimental results were obtained in the time-resolved EPR for E' centres in quartz.
We study the coherent dynamics of relaxing qubits driven by a bichromatic radiation in the dispersive regime, when detuning of the frequency ωrfω_{rf}ωrf​ of a longitudinal radiofrequency field from the Rabi frequency ω1ω_{1}ω1​ in a transverse microwave field is comparable in magnitude to ωrfω_{rf}ωrf​ and ω1ω_{1}ω1​. We analytically describe this regime beyond the rotating wave approximation and find that the dominant feature of dynamics of qubits is the shift of the Rabi frequency caused by the dynamical Zeeman and Bloch-Siegert-like effects. These fundamental effects can be experimentally separated because, unlike the Bloch-Siegert effect, the dynamical Zeeman effect depends on the detuning sign. Our theoretical results are in a good agreement with the experimental data obtained in pulse EPR for the E1′E'_{1}E1′​ centers in crystalline quartz.
We study Rabi oscillations of the second-order Raman transition realized on dressed states of a qubit excited by an amplitude-modulated microwave field. The co-rotating component of the ultrastrong low-frequency modulation field excites virtual multiple photon processes between the dressed states and forms the Rabi frequency in the so-called rotating wave approximation. The counter-rotating modulation component also gives a significant contribution to the Rabi frequency owing to the Bloch-Siegert effect. It is shown that for properly chosen parameters of the modulation field and qubit, the Rabi oscillations in the rotating wave approximation vanish due to destructive interference of multiple photon processes. In this case the Rabi oscillation results exclusively from the Bloch-Siegert effect and is directly observed in the time-resolved coherent dynamics as the Bloch-Siegert oscillation. Correspondingly, in Fourier spectra of the coherent response, triplets are transformed into doublets with the splitting between the lines equal to twice the Bloch-Siegert shift. We demonstrate these features by calculations of the qubit’s evolution in the conditions of experiments with a NV center in diamond, where Raman transitions were observed. The direct observation of the Bloch-Siegert oscillation offers new possibilities for studying driven quantum systems in the ultastrong regime.
The geometric phase in the dynamics of a spin qubit driven by transverse microwave (MW) and longitudinal radiofrequency (RF) fields is studied. The phase acquired by the qubit during the full period of the "slow" RF field manifests in the shift of Rabi frequency ω_{1} of a spin qubit in the MW field. We find out that, for a linearly polarized RF field, this shift does not vanish at the second and higher even orders in the adiabaticity parameter ω_{rf} / ω_{1}, where ω_{rf} is the RF frequency. As a result, the adiabatic (Berry) phases for the rotating and counter-rotating RF components compensate each other, and only the higher-order geometric phase is observed. We experimentally identify that phase in the frequency shift of the Rabi oscillations detected by a time-resolved electron paramagnetic resonance.
We study Rabi oscillations of the second-order Raman transition realized on dressed states of a qubit excited by an amplitude-modulated microwave field.The co-rotating component of the ultrastrong low-frequency modulation field excites virtual multiple photon processes between the dressed states and forms the Rabi frequency in the so-called rotating wave approximation (RWA).The counter-rotating modulation component also gives a significant contribution to the Rabi frequency owing to the Bloch-Siegert effect.It is shown that for properly chosen parameters of the modulation field and qubit, the Rabi oscillations in the RWA vanish due to destructive interference of multiple photon processes.In this case the Rabi oscillation results exclusively from the Bloch-Siegert effect and is directly observed in the time-resolved coherent dynamics as the Bloch-Siegert oscillation.Correspondingly, in Fourier spectra of the coherent response, triplets are transformed into doublets with the splitting between the lines equal to twice the Bloch-Siegert shift.We demonstrate these features by calculations of the qubit's evolution in the conditions of experiments with a NV center in diamond, where Raman transitions were observed.The direct observation of the Bloch-Siegert oscillation offers new possibilities for studying driven quantum systems in the ultastrong regime.
Abstract We study the coherent dynamics of a qubit excited by an amplitude-modulated electromagnetic field under the Rabi resonance when the frequency of the low-frequency modulation field matches the Rabi frequency in the high-frequency field. Due to the destructive interference of multiple photon processes within the ultrastrong coupling between the qubit and the low-frequency driving field, Rabi oscillations result exclusively from the Bloch–Siegert effect. It is directly observed in the time-resolved coherent dynamics as the Bloch–Siegert oscillation. In this case, triplets in the Fourier spectra of the coherent response are transformed into doublets with the splitting between the lines equal to twice the Bloch–Siegert shift. These unusual properties are demonstrated in the conditions of experiments with a nitrogen vacancy center in the diamond.
In the framework of the nonsecular perturbation theory based on the Bogoliubov averaging method, the coherent dynamics of multiphoton Raman transitions in a two-level spin system driven by an amplitude-modulated microwave field is studied. Closed-form expressions for the Rabi frequencies of these transitions are obtained beyond the rotating-wave approximation for the low-frequency driving component. It is shown that spin states dressed by the high-frequency component of the driving field are shifted due to the Bloch-Siegert-like effect caused by antiresonant interactions with the strong low-frequency driving. We predict that with increasing the order of the Raman transition, the Rabi frequency decreases and the contribution of the Bloch-Siegert shift to this frequency becomes dominant. It is found that the amplitude and phase of the Rabi oscillations strongly depend on the initial phase of the low-frequency field as well as on detuning from multiphoton resonance. The recent experimental data for the second- and third-order Raman transitions observed for the nitrogen-vacancy center in diamond (Z. Shu et al., arXiv:1804.10492) are well described in the frame of our approach. Our results may provide new possibilities for coherent control of quantum systems.